On adjunctions for Fourier-Mukai transforms

Abstract

We show that the adjunction counits of a Fourier-Mukai transform $\Phi$ from $D(X_1)$ to $D(X_2)$ arise from maps of the kernels of the corresponding Fourier-Mukai transforms. In a very general setting of proper separable schemes of finite type over a field we write down these maps of kernels explicitly -- facilitating the computation of the twist (the cone of an adjunction counit) of $\Phi$. We also give another description of these maps, better suited to computing cones if the kernel of $\Phi$ is a pushforward from a closed subscheme $Z$ of $X_1\times X_2$. Moreover, we show that we can replace the condition of properness of the ambient spaces $X_1$ and $X_2$ by that of $Z$ being proper over them and still have this description apply as is. This can be used, for instance, to compute spherical twists on non-proper varieties directly and in full generality.

36 pages; v3: Substantially rewritten. Main results strengthened. Includes two new sections - Section 2, a primer on derived categories which everyone should read, and Appendix A, which no one ever should. Final version, to appear in Adv. in Math

References in corpus (1)

• Nagata's embedding theorem

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• New derived autoequivalences of Hilbert schemes and generalised Kummer varieties
• ${\mathbb{P}}^n$-functors
• Orthogonally spherical objects and spherical fibrations
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• Perverse schobers and the Algebra of the Infrared
• Spherical adjunctions of stable $\infty$-categories and the relative S-construction
• Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups
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